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We were given two dice to roll. One is black with six sides. The other is white with four sides. So a four-sided die is kind of a pyramid-looking thing. It has exactly four sides. It would be a pyramid with a triangular base. For a given roll, what is the probability that the dice add up to 10? If it helps, you may select the matching outcomes below. Your selections aren't checked with your answer. So this might look a little bit strange at first. You're like, what is this triangle? This triangle is the face of the four-sided die that is facing you. So this is the roll of the four-sided die. So this is a roll of 1 and 1. So these clearly are not adding up to 10. So let's think about all the ones that add up to 10. So this adds up to 2. This adds up to 3. This adds up to 4. This adds up to 5. Let's see what actually adds up to 10. So if we go all the way-- so clearly, if we get a 4 and a 6, that's going to add up to 10. And are there any other possibilities that add up to 10? Well, the highest possible roll I can get on a four-sided die is a 4. So that's my best scenario. That's the highest possible roll. And even if I get that, I still need the highest possible roll on the six-sided die. So this is the only scenario that adds up to 10. You can look at any of these other ones. They'll add up to something less than 10. So there's one possibility that satisfies our constraints. The dice add up to 10. And how many total equally likely possibilities were there? Let's see. This grid shows all the equally likely possibilities. This is 1, 2, 3, 4 times 1, 2, 3, 4, 5, 6. There are 24 equally likely possibilities, and that comes from you have 6 possibilities from the six-sided die times 4 possibilities from the four-sided die gives you 24 equally likely possibilities. Only one satisfies the constraint that the dice add up to 10. Let's do a few more of these. So you're given two six-sided dice to roll. For a given roll, what is the probability that the dice add to 6? So here we have the tops of both of my dice. And now what are all the scenarios where the dice add up to 6? Well, let's think about this. If I have a 1 and a 5, that's going to add up to 6. In this entire column, that's the only way that I add up to 6. Right over here, let's see. If I have a 2 on the black die, then I'm going to need a 4 on the white die to add up to 6. If I have a 3 on the black die, I'm going to need a 3 on the white die to add up to 6. If I have a 4 on the black die, I'm going to need a 2 on the white die. If I have a 5 on the black die, I'm going to need a 1 on the white die. And then if I have 6 six on the black die, it's actually impossible that the sum will be exactly 6. So there's no zero on the white die. So none of these meet the constraint. So here I have one, two, three, four, five possibilities where my dice add up to 6. So I have five possibilities out of a total of 36 equally likely ones. This is a six by six grid, and it comes from six possibilities for one die times six possibilities for the other die. So 5 possibilities satisfy my constraint out of a total of 36. Let's do one more. This is actually a lot of fun. You're given two six-sided dice to roll. For a given roll, what is the probability at least one die is a six? So let's see. So we just have to look for all of the situations where at least one of the dice is a six. So that one, that one. Every time I see a six, I should just click on that. That one. That one. And then obviously all of these scenarios right over here. So these are all of the scenarios where at least one of the dice is a six. Here, of course, is where both of the dice are a six. So how many scenarios are there? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So there are 11 possibilities out of a total of 36 equally likely ones. Remember, 6 times 6. There's 36 equally likely outcomes. So let's check our answer. And we got it right.